On a modification of Chebyshev's method (Q5906436)

Let (1) \(P(x) = \sum^ N_{s = 0} a_ s \varphi_ s(x)\) be a generalized polynomial by the Chebyshev system of basis functions \(\{\varphi_ s(x)\}^ N_{s = 0}\) with real and complex coefficients \(\{a_ s\}^ N_{s = 0}\) having only isolated zeros \(x_ 1, \dots, x_ N\). For the simultaneous calculation of all zeros of the polynomial (1) the author deduces from the classical Chebyshev method a new method \[ x_ i^{(k + 1)} = x_ i^{(k)} - N \bigl( X_ i^{(k)} \bigr) \;\left[ 1 + {1 \over 2} N \bigl( x_ i^{(k)} \bigr) {Q_ k''(x_ i^{(k)}) \over Q_ k' (x_ i^{(k)})} \right] \tag{2} \] where \(N (x_ i^{(k)}) = P(x_ i^{(k)})/P'(x_ i^{(k)})\), \(Q_ k (x)\) is a polynomial by the same system \(\{\varphi_ s\}^ N_{s = 0}\) of the basis functions and the zeros of \(Q_ k (x)\) are the \(k\)-th approximations \(x_ 1^{(k)}, \dots, x_ N^{(k)}\) to the zeros \(x_ 1, \dots, x_ N\) respectively. He proves that the process (2) converges cubically like the Chebyshev method. Numerical experience with the Chebyshev method and (2) indicates that the last one has a larger domain of convergence than that of the Chebyshev method.